Near-field Acoustical Holography (NAH) is a very useful tool for 3D visualization of sound radiation and for precise noise source localization based on measurements over a surface near the sound source. Its ability to reconstruct also the evanescent wave components ensures a very high spatial resolution.
A known Near-field Acoustical Holography method is based on regular-grid measurements across a level surface in a separable coordinate system, allowing the NAH calculation to be performed by spatial Discrete Fourier Transform (DFT), see e.g. E. G. Williams, J. D. Maynard, and E. J. Skudrzyk, “Sound source reconstruction using a microphone array,” J. Acoust. Soc. Am. 68, 340-344 (1980). Due to the use of DFT, the processing is very fast, but a side effect of using the DFT includes severe spatial windowing effects unless the measurement area fully covers the areas with high sound pressure. In some cases this requirement on the measurement area cannot be fulfilled, and in many cases the necessary size becomes prohibitively large.
A set of techniques have been proposed to reduce the spatial windowing effects, while still maintaining the DFT spatial processing but at the cost of an increased complexity and computational demands, see e.g. J. Hald, “Reduction of spatial windowing effects in acoustical holography,” Proceedings of Inter-Noise 1994. Typically an iterative procedure is first used to extrapolate the measured sound pressure outside the measured area, followed by application of a DFT based holography method on the extended data window.
Other methods have been proposed that seek to avoid the use of spatial DFT and to provide a reduction in the required measurement area.
One such method is the Helmholtz' Equation Least Squares (HELS) method which uses a local model of the sound field in terms of spherical wave functions, see e.g. U.S. Pat. No. 6,615,143, Z. Wang and S. F. Wu, “Helmholtz equation-least-squares method for reconstructing the acoustic pressure field,” J. Acoust. Soc. Am, 102(4), 2020-2032 (1997); or S. F. Wu, “On reconstruction of acoustic fields using the Helmholtz equation-least-squares method,” J. Acoust. Soc. Am, 107, 2511-2522 (2000). However, since only spherical wave functions with a common origin are used to represent the sound field, errors will be introduced in the sound field reconstruction on the source surface, unless the source surface is also spherical and centered in the same origin. Another drawback of the above prior art method is that it requires a large number of measurement positions to obtain a sufficiently accurate model. A third drawback is that traditional regularization methods like Tikhonov regularization do not work properly. Instead, the above prior art method applies a computationally expensive iterative search for an optimal truncation of the spherical wave expansion combined with a least squares solution without regularization.
Another previously proposed method is the Statistically Optimized Near-field Acoustic Holography (SONAH) method disclosed in R. Steiner and J. Hald, “Near-field Acoustical Holography without the errors and limitations caused by the use of spatial DFT,” Intern. J. Acoust. Vib. 6, 83-89 (2001). However, this prior art method is based on plane wave functions and does not allow for accurate reconstruction of the sound field on non-planar source surfaces.